On the first day to sign up, the “NY State of Health” Web site wouldn’t let New Yorkers connect for hours, and then asked everyone to give it a rest.

“Due to overwhelming interest in the NY State of Health — including 2 million visits in the first 2 hours of the site launch — the health exchange is currently having log-in issues,” a banner message read.

In Tribeca, Shireen Imani, 33, struggled fruitlessly all day to purchase a policy and avoid the penalty imposed under ObamaCare’s “individual mandate.”

Although the penalty is capped at $95 for lower-income workers, Imani — who makes more than $200,000 a year running a nurse-recruiting business — could get soaked for 1 percent of her income, or more than $2,000.

After hours of attempts, Imani was finally able to log in and create an account, only to then get booted off the site by an “Internal Server Error.”

At 5 p.m., she gave up after a customer-service rep told her: “Even us, we have a few hundred employees and we can’t log in.”

“I’m bummed. I’ve been trying this for six hours,” Imani said, shaking her head.

Exchanges in 36 states being run by the federal government were plagued by similar problems, with error messages posted on at least 25 of them just minutes after the system went live at 8 a.m.

Obama acknowledged the headache-inducing holdups but defended the rollout during a Rose Garden news conference.

“Like every new product, [there are] gonna be some glitches that we can fix,” he said.